Blender Scale and Unit problem, how to fix?
As this answer suggests:
How can I make the default size of new objects 1 x 1?
You can, along changing meters to centimeters, also change the unit scale to 0.01 (as 1 cm is 0.01 m)

This will probably solve all your problems not just huge cubes being added, but you may find it important, that when using proportional editing, there's a circle of influence visible, which you can scale by using your mouse wheel. Due to your scale problems, this circle could be so big, that it was entirely outside of the viewport, so you should see it if you zoom out considerably before moving / rotating / scaling or if you scroll your mouse wheel up for a while:

The rate at which this circle's radius changes will depend on its current radius (the change gets slower as the circle gets smaller) and your zoom (so the change of the circle's size depends on its size in viewport). However, you can still decrease the change for a finer control, by holding SHIFT key.
Likewise, holding SHIFT will decrease step size in input fields:




no SHIFT
SHIFT